Chordal embeddings of planar graphs - LaBRI

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Chordal embeddings of planar graphs - LaBRI

Chordal embeddings of planar graphs

V. Bouchitté 1 , F. Mazoit 1 , and I. Todinca 2

1 LIP-École Normale Supérieure de Lyon

46 Allée d’Italie, 69364 Lyon Cedex 07, France

email: {Vincent.Bouchitte,Frederic.Mazoit}@ens-lyon.fr

2 LIFO-Université d’Orléans

BP 6759, 45067 Orleans Cedex 2, France

email: Ioan.Todinca@lifo.univ-orleans.fr

Abstract. Robertson and Seymour conjectured that the treewidth of a planar

graph and the treewidth of its geometric dual differ by at most one.

Lapoire solved the conjecture in the affirmative, using algebraic techniques.

We give here a much shorter proof of this result.

1 Introduction

The notions of treewidth and tree decomposition of a graph have been introduced by

Robertson and Seymour in [14] for their study of minors of graphs. These notions

have been intensively investigated for algorithmical purposes since many NP-hard

problems become polynomial and even linear when restricted to classes of graphs

with bounded treewidth.

Robertson and Seymour conjectured in [13] that the treewidth of a planar graph

and the treewidth of its geometric dual differ by at most one. Lapoire [11] solved

this conjecture in the affirmative, in fact he proved a more general result. In order

to prove his result, Lapoire worked on hypermaps and introduced the notion of

splitting of hypermaps, his approach is essentially an algebraic one.

Computing the treewidth of an arbitrary graph is NP-hard. Nevertheless, the

treewidth can be computed in polynomial time for several well-known classes of

graphs, for example chordal bipartite graphs [9], circle and circular-arc graphs [8]

[16], permutation graphs [2] and weakly triangulated graphs [3]. Actually all these

classes of graphs have a polynomial number of minimal separators, we proved in

[4] that we can compute, in polynomial time, the treewidth of a graph in any class

having a polynomial number of minimal separators.

For classes of graphs having an exponential number of minimal separators, we

know very few, for instance the problem remains NP-hard on AT-free graphs [1] and

it is polynomial for rectangular grids. Maybe the most challenging open problem is

the computation of the treewidth for planar graphs. In [15], Seymour and Thomas

gave a polynomial time algorithm that approximate the treewidth of planar graph

within a factor of 3 2 .

In this paper, we give a new approach to tackle the problem of the treewidth

computation for planar graphs. First, we recall how to obtain minimal chordal

embeddings of graphs by completing some families of minimal separators. Secondly,

we show that we can interpret minimal separators of planar graphs as Jordan curves

of the plane. Then, we study the structure of Jordan curves that give a minimal

triangulation of the graph. Next, given a family of curves of the plane, we show how

to build a minimal triangulation of the geometric dual of the graph. Finally, given

an optimal triangulation w.r.t. treewidth of the initial graph, we give a triangulation

of the dual graph whose maximal cliquesize is no more than the maximal cliquesize

of the original graph plus one. So, we get a new proof of the conjecture of Robertson

and Seymour which is much simpler than the proof of Lapoire.


2 Preliminaries

Throughout this paper we consider simple, finite, undirected graphs.

A graphG=(V,E) is planar if it can be drawn in the plane such that no two

edges meet in a point other than a common end. The plane will be denoted byΣ. A

plane graphG = (V,E) is a drawing of a planar graph. That is, each vertexv∈V

is a point ofΣ, each edgee∈E is a curve between two vertices, distinct edges have

distinct sets of endpoints and the interior of an edge contains no point of another

edge. A face of the plane graphG is a region ofΣ\G.F(G) denotes the set of faces

ofG. Sometimes we will also use plane multigraphs, i.e. we allow loops and multiple

edges.

LetG=(V,E) a plane graph. The dualG ∗ = (F,E ∗ ) ofGis a plane multigraph

obtained in the following way: for each face ofG, we place a pointf into the face,

and these points form the vertex set ofG ∗ . For each edgeeofG, we link the two

vertices ofG ∗ corresponding to faces incident toeinG, by an edgee ∗ crossinge; if

e is incident with only one face, thene ∗ is a loop.

A graphH is chordal (or triangulated) if every cycle of length at least four has

a chord. A triangulation of a graphG=(V,E) is a chordal graphH = (V,E ′ )

such thatE⊆E ′ .H is a minimal triangulation if for any intermediate setE ′′ with

E⊆E ′′ ⊂E ′ , the graph (V,E ′′ ) is not triangulated. We point out that in this

paper, a triangulation of a planar graphGwill always mean a chordal embedding

ofG. Thus, a triangulation ofGis clearly not equivalent to a planar triangulation

(that is, a planar supergraph such that each face of the supergraph is a triangle) of

G.

Definition 1. LetG = (V,E) be a graph. The treewidth ofG, denoted bytw(G),

is the minimum, over all triangulationsH ofG, ofω(H)−1, whereω(H) is the

the maximum cliquesize ofH. The treewidth of a multigraph is the treewidth of the

corresponding simple graph.

The aim of this paper is to prove the following assertion, stated by Robertson

and Seymour in [13]:

Problem 1. For any plane graphG = (V,E),

tw(G ∗ )≤tw(G)≤tw(G ∗ ) + 1.

We say that a graphG ′ is a minor of a graphG if we can obtainG ′ fromGby

repeatedly using the following operations: vertex deletion, edge deletion and edge

contraction. Kuratowski’s theorem states that a graphGis planar if and only if the

graphsK 3,3 andK 5 are not minors ofG. It is well-known that ifG ′ is a minor of

G, then tw(G ′ )≤tw(G). We reffer to [5] for more details on these results.

When we compute the treewidth of a graphG, we are searching for a triangulation

ofGwith smallest cliquesize, so we can restrict our work to minimal triangulations.

We need a characterization of the minimal triangulations of a graph, using

the notion of minimal separator.

A subsetS⊆V is ana,b-separator for two nonadjacent verticesa,b∈V if the

removal ofS from the graph separatesaandbin different connected components.

S is a minimala,b-separator if no proper subset ofS separatesaandb. We say

thatS is a minimal separator ofGif there are two verticesaandbsuch thatS is

a minimala,b-separator. Notice that a minimal separator can be strictly included

into another. We denote by∆ G the set of all minimal separators ofG.

LetGbe a graph andS be a minimal separator ofG. We denote byC G (S) the

set of connected components ofG\S. A componentC∈C G (S) is full if every vertex

ofS is adjacent to some vertex ofC. For the following lemma, we refer to [7].

2


Lemma 1. A setS of vertices ofGis a minimala,b-separator if and only ifaand

b are in different full components ofS.

Definition 2. Two separatorsS andT cross, denoted byS♯T , if there are some

distinct componentsC andDofG\T such thatS intersects both of them. IfS and

T do not cross, they are called parallel, denoted byS‖T .

It is easy to prove that these relations are symmetric.

LetS∈∆ G be a minimal separator. We denote byG S the graph obtained

fromGby completingS, i.e. by adding an edge between every pair of non-adjacent

vertices ofS. IfΓ⊆∆ G is a set of separators ofG,G Γ is the graph obtained by

completing all the separators ofΓ . The results of [10], concluded in [12], establish

a strong relation between the minimal triangulations of a graph and its minimal

separators.

Theorem 1. LetΓ∈∆ G be a maximal set of pairwise parallel separators ofG.

ThenH=G Γ is a minimal triangulation ofGand∆ H =Γ .

LetH be a minimal triangulation of a graphG. Then∆ H is a maximal set of

pairwise parallel separators ofGandH =G ∆H . Moreover, for eachS∈∆ H , the

connected components ofH\S are exactly the connected components ofG\S.

In other terms, every minimal triangulation of a graphGis obtained by considering

a maximal setΓ of pairwise parallel separators ofGand completing the

separators ofΓ . The minimal separators of the triangulation are exactly the elements

ofΓ .

3 Minimal separators as curves

We show in this section that, in plane graphs, we can associate to each minimal

separatorS a Jordan curve such that, ifS separates two vertices of the graph, then

the curve separates the corresponding points in the plane.

Definition 3. LetG = (V,E) be a planar graph. We fix a plane embedding ofG.

LetF be the set of faces of this embedding. The intermediate graphG I = (V∪F,E I )

has vertex setV∪F. We place an edge inG I between an original vertexv∈V and

a face-vertexf∈F whenever the corresponding vertex and face are incident inG.

Proposition 1. LetGbe a 2-connected plane graph. Then the intermediate graph

G I is also 2-connected.

Proof. Let us prove that, for any couple of original verticesxandy ofG I and for

any face or original vertexa, there is anx,y-path inG I avoidinga. Letµ=[x =

v 1 ,v 2 ,...,v p =y] anx,y-path ofG. Ifa∈V (G), since{a} is not anx,y separator

ofG, we can chooseµsuch thata∉µ. For each edgee i =v i ,v i+1 , 1≤i


Proposition 2. Consider a cycleν ofG I . Its drawing defines a Jordan curve ˜ν in

the plane. Removing ˜ν separates the plane into two regions. If both regions contain

at least one original vertex, then the original vertices ofν form a separator ofG.

Proof. Letxandybe two original vertices, separated by the curve ˜ν in the plane.

Clearly, no edge ofGcrosses an edge ofG I , and therefore no edge ofGcrosses the

curve ˜ν. Every pathµconnectingxandy inGintersects ˜ν, soµhas a vertex inν.

It follows thatν∩V is ax,y-separator ofG. ⊓⊔

Proposition 3. LetS be a minimal separator of a 2-connected plane graphGand

C be a full component associated toS. ThenScorresponds to an elementary cycle

ν S (C) ofG I , of the same original vertices and of equal number of face-vertices in

G I , such thatG I \ν S (C) has at least two connected components. Moreover, the

original vertices of one of these components are exactly the vertices ofC.

Proof. LetC be a full component associated toS, letG C be formed by contracting

C into a supervertex, and letS ′ be the set of faces and vertices adjacent inG C to

the contracted supervertex. ThenS ′ is neighborhood of the supervertex inG C I , so

it has the structure of a cycle inG C I and therefore inG I. This cycle will be denoted

ν S (C). SinceC is a full component associated toS inG, we have thatS=N G (C),

so the original vertices ofS ′ are exactly vertices ofS. The cycle separatesC from

V\{S∪C} inG I .

⊓⊔

The cycleν S (C) defined in the previous proposition will be called the cycle

associated toS andC, close toC.

Remark 1. Any cycleν ofG I forms a Jordan curve in the plane. We denote ˜ν this

curve. Removing ˜ν separates the plane into two open regions. Consider the cycle

ν S (C) ofG I associated to a minimal separatorS and a full componentC ofG\S,

close toC. Then one of the regions defined by ˜ν S (C) contains all the vertices ofC

and the other contains all the vertices ofV\ (S∪C).

4 Some technical lemmas

In the next section we show how to associate to each minimal separatorS of the

3-connected plane graphGaunique cycle ofG I having good separation properties.

We group here some technical lemmas that will be used in the next sections.

Lemma 2. LetGbe a 3-connected planar graph andS be a minimal separator of

G. ThenG\S has exactly two connected components.

Proof. By Lemma 1, there are two distinct full componentsC 1 andC 2 associated to

S. Suppose there is another componentC 3 ofG\S and letS 3 =N(C 3 ). Clearly,S 3 is

a separator ofG, so|S 3 |≥3. Letx 1 ,x 2 ,x 3 be three distinct vertices ofS 3 . Consider

the plane graphG ′ obtained fromGby contracting each componentC 1 ,C 2 andC 3

into a supervertex. The three supervertices are adjacent inG ′ tox 1 ,x 2 ,x 3 , soG ′

contains a subgraph isomorphic toK 3,3 – contradicting Kuratowski’s theorem. ⊓⊔

Proposition 4. LetS be a minimal separator of a 3-connected planar graphG.

ThenS is also an inclusion minimal separator ofG.

Proof. Suppose there is a separatorT ofGsuch thatT⊂S. There is a connected

componentC ofG\T such thatC∩S =∅. Indeed, ifS intersects each component

ofG\T , thenS andT cross, and since the crossing relation is symmetricT must

intersect two connected components ofG\S, contradictingT⊂S. SinceS∩C =∅

andT ⊂S,C is also a connected component ofG\S. By Lemma 1, there are

two full componentsD 1 ,D 2 associated toS. Notice thatC is not a full component

associated toS, becauseN(C) =T⊂S. It follows thatD 1 ,D 2 andC are three

distinct components associated toS inG, contradicting Lemma 2. ⊓⊔

4


Lemma 3. LetGbe a plane graph andν be a cycle ofG I such that ˜ν separates two

original verticesaandbin the plane. Consider two verticesxandyofν. Suppose

there is a pathµfromxtoy inG I , such thata,b∉µ andµdoes not intersect the

cycleν except inxandy.

The verticesxandy splitν into twox,y-paths ofG I , denotedµ 1 andµ 2 . Consider

the cyclesν 1 (respectivelyν 2 ) ofG i formed by the pathsµandµ 1 (respectively

µ andµ 2 ). Then ˜ν 1 or ˜ν 2 separate two original vertices in the plane.

Proof. LetR 1 ,R 2 be the two regions obtained by removing ˜ν from the plane. By

hypothesis, bothR 1 andR 2 contain original vertices, saya∈R 1 andb∈R 2 .

Suppose w.l.o.g. that the pathµis contained inR 1 ∪{x,y}. Then the drawing ofµ

splitsR 1 into two regions:R 1 ′ , bordered by the curve ˜ν 1, andR 1 ′′,

bordered by ˜ν 2.

Ifa∈R 1 ′ then ˜ν 1 separatesaandbin the plane, otherwisea∈R 1 ′′ so ˜ν 2 separates

a andbin the plane.

⊓⊔

Lemma 4. LetS be a minimal separator of a 3-connected plane graphG. Consider

a cyleν S ofG I such that the original vertices ofν S are the elements ofS. Suppose

that ˜ν S separates in the plane two original vertices ofG.

If two original vertices ofS are at distance two inG I (i.e. they are incident to

a same face ofG), these vertices are also at distance two on the cycleν S .

Proof. Letν S = [v 1 ,f 1 ,...,v p ,f p ], wherev i (respectivelyf i ) are the original (respectively

face) vertices ofν S . The conclusion is obvious ifp≤3. Suppose there

are two verticesx,y∈S at distance two inG I , but not inν S . W.l.o.g., we suppose

x =v 1 andy=v i , 3≤i≤p−2. Letf be a face vertex adjacent tov 1 andv i in

G I .

Iff ∉ν S (C), we apply Lemma 3 with cycleν S and path [v 1 ,f,v i ], so one

of the cyclesν 1 = [v 1 ,f 1 ,v 2 ,f 2 ,...,v i ,f] orν 2 = [v 1 ,f,v i ,f i ,v i+1 ,f i+1 ,...,v p ,f p ]

separates two original vertices in the plane (see figure 1a). By proposition 2, the

original vertices ofν 1 orν 2 form a separatorT inG. ButT is strictly contained in

S, contradicting proposition 4.

v 1 f 1

f 1

v 2

v 2

v 1

f

f

v i+1 f i+1

˜ν S

f i v

v ˜ν S

i+1 i f i

(a)

(b)

Fig. 1. Proof of Lemma 4

The casef∈ν S is very similar. There is somej, 1≤j≤psuch thatf =f j .

Sincev 1 andv i are not at distance two onν S , we have thatj∉{1,p} orj∉

{i−1,i + 1}. Suppose w.l.o.g. thatf is not consecutive tov 1 on the cycleν S . We

apply Lemma 3 with cycleν S and path [v 1 ,f]. We obtain that one of the cycles

ν 1 = [v 1 ,f 1 ,...,v j ,f j ] orν 2 = [v 1 ,f j ,v j+1 ,f j+1 ,...,v p ,f p ] (see figure 1b) separates

two original vertices ofG, so the original vertices ofν 1 orν 2 form a separatorT of

G. In both cases,T⊂S, contradicting proposition 4. ⊓⊔

Lemma 5. LetG = (V,E) be a plane graph andx,y∈V such that at least three

faces are incident to bothxandy. Then{x,y} is a separator ofG.

5


Proof. Letf 1 ,f 2 ,f 3 be three faces incident to bothxandy. Consider the three

pathsµ i = [x,f i ,y], 1≤i≤3 of the intermediate graphG I . The drawings of these

paths split the plane into three regions:R 1 bordered by the cycleν 1 = [x,f 2 ,y,f 3 ],

R 2 bordered byν 2 = [x,f 1 ,y,f 3 ] andR 3 bordered byν 3 = [x,f 1 ,y,f 2 ]. We show

that at least two of the three regions contain one or more original vertices.

Suppose thatR 2 andR 3 do not contain original vertices. In the graphG, each

face is incident to at least three vertices, sof 1 has a neighborz, different fromx

andy. The edgef 1 z ofG i does not cross any of the pathsµ 1 ,µ 2 ,µ 3 , soz is in one

of the regionsR 2 orR 3 , incident tof 1 . This contradicts our assumption thatR 2

andR 3 do not contain original vertices.

We proved that at least two of the three regionsR 1 ,R 2 ,R 3 – sayR 1 andR 2

– contain original vertices. Then ˜ν 1 , the Jordan curve borderingR 1 , separates two

original vertices ofG. By proposition 2, the original vertices ofµ 1 , namely{x,y},

form a separator ofG.

⊓⊔

Lemma 6. LetG = (V,E) be a 3-connected plane graph andx,y∈V .

1. Ifxy∈E, there are exactly two faces incident to bothxandy.

2. Ifxy∉E, there is at most one face ofGincident to bothxandy.

Proof. The graphGis 3-connected, so by Lemma 5 there are at most two faces

incident to bothxandy.

The first statement is obvious since the edgexy is incident to two faces ofG.

For the second statement, suppose there are two facesf 1 andf 2 incident toxandy.

Consider the plane graphG ′ obtained fromGby adding the edgexy, drawn in the

facef 1 . Then the facef 1 ofGis splitted into two facesf 1 ′ andf′′ 1 , both incident to

x andy inG ′ . So, inG ′ , the three facesf 1 ′,f′′

1 andf 2 are incident to bothxandy.

ButG ′ is clearly a 3-connected planar graph, and by Lemma 5 we have that{x,y}

is a separator ofG ′ – a contradiction. ⊓⊔

Proposition 5. LetGbe a 3-connected plane graph. Consider two cyclesν andν ′

ofG I , such thatν andν ′ only differ by their face vertices. Then ˜ν separates two

original verticesaandbin the plane if and only if ˜ν ′ also separatesaandbin the

plane.

Proof. It is sufficient to prove our statement for two cycles that only differ by one

face vertex, sayν = [v 1 ,f 1 ,v 2 ,f 2 ,...,v p ,f p ] andν ′ = [v 1 ,f 1,v ′ 2 ,f 2 ,...,v p ,f p ], such

thatf 1 ≠f 1 ′. Sincev 1 andv 2 are incident to bothf 1 andf 1 ′ inG, it comes by

Lemma 6 thatv 1 andv 2 are adjacent inG. Thus,f 1 andf 1 ′ are the faces incident

inGto the edgee =v 1 v 2 .

Consider the cycleν ′′ = [v 1 ,f 1 ,v 2 ,f 1 ′] ofG I and letRbe the region bordered

by ˜ν ′′ and containing the interior of the edgee. Clearly, the regionRcontains no

original or face vertex ofG I .

LetR 1 ,R 2 be the two regions obtained by removing ˜ν from the plane. Suppose

that the edgee, and thus the facef 1, ′ is inR 2 . Then the regions obtained by removing

˜ν ′ from the plane are exactlyR 1 ′ =R 1∪R∪[v 1 ,f 1 ,v 2 ] andR 2 ′ =R 2\R\[v 1 ,f 1 ′,v 2].

SinceRcontains no original vertices, the original vertices ofR 1 ′ (respectivelyR 2)


are the original vertices ofR 1 (respectivelyR 2 ). ⊓⊔

Lemma 7 ([5], proposition 4.2.10). LetGbe a 3-connected plane graph. For

any facef, the set of vertices incident tof do not form a separator ofG.

Lemma 8. LetGbe a 3-connected plane graph andS be a minimal separator ofG.

Then each face ofGis incident to at most two vertices ofS.

6


Proof. Suppose there are three verticesx,y,z ofS incident to a same facef. Let

C be a full component associated toSinGandν S (C) be the cycle associated

toSinG I , close toC. Consider first the case|S|≥4, so there is some vertex

t∈S\{x,y,z}. Suppose w.l.o.g. thatν S (C) encountersx,y,z andtin this order.

Thenxandz are not at distance 2 on the cycleν S (C), contradicting Lemma 4.

If|S| = 3, letT be the set of vertices incident tof, soS⊆T . Thus,T is a

separator ofG, contradicting Lemma 7. ⊓⊔

5 Minimal separators in 3-connected planar graphs

Consider a minimal separatorS ofGand two full componentsC andDassociated

toS. We can associate toS two cycles ofG I , namelyν S (C) andν S (D), closed to

C, respectivelyD. In general, the two cycles are distinct, although they represent

for us the same minimal separatorS. In the case of 3-connected planar graphs, we

slightly modify the construction of proposition 3 in order to obtain a unique cycle

representingS inG I .

LetGbe a 3-connected plane graph. Consider two original verticesxandy

situated at distance two inG I . We know thatxany are incident to a same face

inG, but this face is not necessarily unique. For each pair of verticesx,y∈V at

distance two inG I , we fix a unique facef(x,y) ofGincident to bothxandy. Let

ν = [v i ,f 1 ,v 2 ,f 2 ,...,v p ,f p ] be a cycle ofG I , wherev i are the original vertices and

f i are the face vertices ofν. We say that a cycleν is well-formed if, for each pair

of consecutive original verticesv i ,v i+1 ofν we havef i =f(v i ,v i+1 ) (1≤i≤p,

v p+1 =v 1 ).

Given a minimal separatorS ofGwe construct a unique well-formed cycleν S

associated toS as follows.

LetC,D be the full components associateds toS inGand letν S (C) = [v 1 ,f 1 ,v 2 ,

f 2 ,...,v p ,f p ] be the cycle associated toS inG I , close toC. We denoteν S ′ (C) =

[v 1 ,f 1 ′,v 2,f 2 ′,...,v p,f p ′] wheref′ i =f(v i,v i+1 )∀i, 1≤i≤p. Notice that∀i,j, 1≤

i


We prove that ifν S crossesν T , thenS crossesT . LetRandR ′ be the regions

ofΣ\˜ν S . We show that at least one original vertex ofν T is inR. Sinceν S crosses

ν T , ˜ν T intersectsR. Thus, an original vertex or a face-vertex ofν T is inR. Suppose

thatν T has no original vertex inRand letf be a face-vertex ofν T ∩R. On the

cycleν T , the face-vertexf is between two original verticesxandx ′ . Notice thatx

is also a vertex ofν S . Indeed,x∉R, andxcannot be inR ′ , because the edgexf

ofG I cannot cross the drawing of the cycleν S . It follows thatx∈ ˜ν S . Soxand

x ′ are both vertices ofν S . Sincexandx ′ are adjacent to a same face-vertex ofG I ,

they are on a same face ofG. By Lemma 4,xandx ′ are at distance two on the

cycleν S , and letf ′ be the face-vertex ofν S betweenxandx ′ . Sinceν S andν T are

well-formed cycles, we havef ′ =f(x,y) =f, sof∈ν S . This contradicts the fact

thatf is in one of the regions ofΣ\˜ν S .

We showed thatν S has original vertices in regionR, and for similar reasons it has

original vertices inR ′ . So ˜ν S separates two original vertices ofν T in the plane, and

by proposition 6S separates these vertices inG. Thus,S crossesT . ⊓⊔

6 Block regions

Let ˜ν be a Jordan curve in the plane. LetRbe one of the regions ofΣ\˜ν. We say

that (˜ν,R) = ˜ν∪R is a one-block region of the plane, bordered by ˜ν.

Definition 5. Let ˜C be a set of curves such that for each ˜ν∈ ˜C, there is a oneblock

region (˜ν,R(˜ν)) containg all the curves of ˜C. We define the region between

the elements of ˜C as

RegBetween( ˜C) =

⋂˜ν∈ ˜C(˜ν,R(˜ν))

We say that the region between the curves of ˜C is bordered by ˜C.

Definition 6. A subsetBR⊆Σ of the plane is a block region if one of the following

holds:

–BR =Σ.

– There is a curve ˜ν such thatBR is a one-block region (˜ν,R).

– There is a set of curves ˜C such thatBR =RegBetween( ˜C).

Remark 2. According to our definition, block regions are always closed sets.

˜µ 2

˜µ 1 ˜µ 3

(a)

(b)

Fig. 2. Block regions

8


Example 1. In figure 2a, we have a block region (in grey) bordered by four Jordan

curves. Figure 2b presents three interior-disjoint paths ˜µ 1 , ˜µ 2 and ˜µ 3 having the

same endpoints. Consider the three curves ˜ν 1 = ˜µ 2 ∪˜µ 3 , ˜ν 2 = ˜µ 1 ∪˜µ 3 and ˜ν 3 = ˜µ 1 ∪˜µ 2 .

Notice that the block-region between ˜ν 1 , ˜ν 2 and ˜ν 3 is exactly the union of the three

paths.

Consider a set ˜C of pairwise parallel Jordan curves of the plane. These curves

split the plane into several block regions. Consider the set of all the block regions

bordered by some elements of ˜C. We are interested by the inclusion-minimal elements

of this set, that we call minimal block regions formed by ˜C. The following proposition

comes directly from the definition of the minimal block-regions:

Proposition 8. Let ˜C be a set of pairwise parallel curves in the planeΣ. A set of

pointsAof the plane are contained in a same minimal block-region formed by ˜C if

and only if for any ˜ν∈ ˜C, there is a one-block region (˜ν,R(˜ν)) containingA.

7 Minimal triangulations ofG

LetGbe a 3-connected planar graph and letH be a minimal triangulation of

G. According to Theorem 1, there is a maximal set of pairwise parallel separators

Γ⊆∆ G such thatH =G Γ . LetC(Γ ) ={ν S |S∈Γ} be the cycles associated to

the minimal separators ofΓ and let ˜C(Γ ) ={˜ν S |S∈Γ} be the curves associated

to these cycles. According to proposition 7, the cycles ofC(Γ ) are pairwise parallel.

Thus, the curves of ˜C(Γ ) split the plane into block regions. We show that any

maximal cliqueΩ ofH corresponds to the original vertices contained in a minimal

block region formed by ˜C(Γ ).

IfBR is a block region, we denote byBR G the vertices ofGcontained inBR.

Theorem 2. LetH =G Γ be a minimal triangulation of a 3-connected planar graph

G.Ω⊆V is a maximal clique ofH if and only if there is a minimal block region

BR formed by ˜C(Γ ) such thatΩ =BR G .

Proof. LetBR be a minimal block region formed by ˜C(Γ ), we show thatΩ=BR G

is a clique ofH. Suppose there are two verticesx,y∈Ω, non adjacent inH. Thus,

there is a minimal separatorS ofH separatingxandyinH. ThenS is also

a minimal separator ofG, separatingxandyinG(cf. Theorem 1). Therefore,

˜ν S ∈ ˜C(Γ ) separatesxandy in the plane, contradicting proposition 8.

LetΩ be a clique ofH. For any minimal separatorS ofH there is a connected

componentC(S) ofH\S such thatΩ⊆S∪C(S). By Theorem 1,S∈Γ andC(S)

is a connected component ofG\S, so we deduce that the points ofΩ are contained

in a same one-block region (˜ν S ,R(˜ν S )) defined by ˜ν S . This holds for eachS∈Γ ,

because the minimal separators ofH are exactly the elements ofΓ . We conclude by

proposition 8 thatΩ is contained in some minimal blockBR formed by ˜C(Γ ). ⊓⊔

8 Triangulations of the dual graphG ∗

LetGbe a plane graph andC be a set of pairwise parallel cycles ofG I . The family

˜C of curves associated to these cycles splits the plane into block regions. LetG ∗ be

the dual ofG. We show in this section how to associate toC a triangulationH(C)

ofG ∗ such that each clique ofH(C) corresponds to the face-vertices contained in

some minimal block-region defined by ˜C.

Definition 7. Consider a planar embedding of the graphG = (V,E) and letG ∗ =

(F,E ∗ ) the dual ofG. LetC be a set of pairwise parallel cycles ofG I . We define the

9


graphH(C) = (F,E H ) with vertex setF, we place an edge between two face-vertices

f andf ′ ofH if and only iff andf ′ are in same a minimal block region defined

by ˜C.

Theorem 3.H(C) is a triangulation ofG ∗ . Moreover, for any cliqueΩ ∗ ofH(C)

there is some minimal block regionBR defined by ˜C such thatΩ ∗ is formed by the

face-vertices contained inBR.

Proof. We show thatH is a supergraph ofG ∗ . Letff ′ be an edge ofG ∗ , clearly no

cycle ofG I crosses the edgeff ′ in the plane. Thus, for any cycle ˜ν of ˜C, ˜ν does not

separate the pointsf andf ′ . By proposition 8,f andf ′ are in a same block region

formed by ˜C, soff ′ is an edge ofH(C).

We prove now thatH(C) is chordal. Suppose there is a chordless cycleν H of

H(C), having at least four vertices. Letf,f ′ be two non-adjacent vertices ofν H . By

proposition 8, there is a curve ˜ν∈ ˜C separating the pointsf andf ′ in the plane.

Consider the two interior-disjoint pathsµ 1 andµ 2 fromf tof ′ inν H . We show

that at least one face-vertex of each of these paths belongs to ˜ν. Letµ 1 = [f =

f 1 ,f 2 ,...,f p =f ′ ].

LetRandR ′ be the regions ofΣ\˜ν containingf, respectivelyf ′ . Letf j the

last point ofµ 1 contained inR, so 1≤j0. Letxandy two vertices ofBR G , by induction hypothesis there

exists a cycleν ′ containingxandyinside∩ k i=2 ( ˜ν i,R( ˜ν i )). Ifν ′ intersectsν 1 in at

most one vertex then the cycleν ′ is insideBR.

10


Otherwise, letµ x (resp.µ y ) be the path ofν ′ that containsx(resp.y) and whose

the only vertices in common withν 1 are its endsx 1 andx 2 (resp.y 1 andy 2 ). If

µ x =µ y then we can completeµ x in a simple cycle that belongs to ( ˜ν 1 ,R( ˜ν 1 )) by

followingν 1 fromx 1 tox 2 . Ifµ x ≠µ y , on the cycleν 1 , the verticesx 1 andx 2 and

the verticesy 1 andy 2 are juxtaposed (see figure 3). There are two disjoint pathsµ 1

andµ 2 ofν 1 whose ends arex 1 andx 2 , respectivelyy 1 andy 2 . The four pathsµ 1 ,

µ 2 ,µ x andµ y form a simple cycle that lies inside (˜ν 1 ,R(˜ν 1 )).

x 1 y 1

µ 1

˜ν

x

y

µ 2

x 2 y 2

˜ν ′

Fig. 3. Cycle inside the block-region

Moreover in both cases, sinceν 1 also lies inside∩ n i=2 ( ˜ν i,R( ˜ν i )) the new cycle

lies inside∩ n i=2 ( ˜ν i,R( ˜ν i )) and so insideBR.

In any case, we can exhibit a cycle insideBR passing throughxandy soBR G

is 2-connected.

⊓⊔

Lemma 10. LeG = (V,E) be a 2-connected planar graph, consider a familyC of

pairwise parallel cycles. LetBR be a block-region defined by a subfamily ˜C ′ of ˜C and

ν∈C ′ .

Either all verticesBR G are onν or there exists a pathµ⊆BR G which intersects

ν only in its extremities.

Proof. Suppose there exists a vertexx∈BR G which is not onν. We know by

Lemma 9 thatBR G is 2-connected. Take two verticesy andz onν, applying Dirac’s

fan lemma toxand{y,z}, we get two disjoint paths except inx,µ y andµ z inBR G ,

connecting respectivelyxtoyandxtoz. Cuttingµ y (resp.µ z ) at the first vertex

y ′ (resp.z ′ ) onν, we obtain two pathsµ y ′ andµ z ′ which intersectν only iny ′ and

z ′ . Then the concatenation ofµ y ′ andµ z ′ is a suitable pathµ. ⊓⊔

Theorem 4. LetG = (V,E) be a 3-connected planar graph. LetC be an inclusion

maximal family of pairwise parallel cycles ofG I . Consider a minimal block-region

BR ofG I defined by ˜C then eitherBR GI =ν orBR GI =ν∪µ, whereν is inC

andµis a path which touchesν only in its ends.

Proof. IfBR GI is not a cycle, we know by Lemma 10 that there exists a pathµ

insideBR GI that intersectsν only on its extremitiesxandy. The verticesxand

y define two subpathsν 1 andν 2 ofν, so we have three cycles contained inBR GI ,

namelyν,µν 1 andµν 2 . These cycles are pairwise parallel and parallel to the cycles

ofC, so by maximality ofC they are inC. These three cycles define a block-region

BR ′ which is exactly ˜ν∪ ˜µ. SinceBR ′ = ˜ν∪ ˜µ⊆BR andBR is a minimal blockregion

we can conclude thatBR ′ =BR. ⊓⊔

11


Theorem 5. LetG = (V,E) be a 3-connected planar graph without loops. Then

tw(G)−1≤tw(G ∗ )≤tw(G) + 1.

Proof. By duality, it is sufficient to prove the second inequality. SinceGis 3-

connected without loops,G,G ∗ and, by proposition 1,G I are 2-connected without

loops. LetC be a family of cycles ofG I that gives a triangulationH ofGwith

ω(H)−1 =tw(G). We completeC into a maximal familyC ′ of pairwise parallel

cycles ofG I .

According to Theorem 3, the familyC ′ defines a triangulationH ∗ ofG ∗ . Let

BR be a minimal block-region with respect to ˜C ′ . By Theorem 4, eitherBR GI =ν

orBR GI =ν∪µ. In the first case, sinceG I is bipartite we have|BR GI ∩V| =

|BR GI ∩V ∗ |. In the later case, the difference between the number of vertices of

G andG ∗ ofBR GI comes fromµ. Once again, sinceG I is bipartite the difference

can be at most one. But each minimal block-region formed by ˜C ′ is contained in a

minimal block-region formed by ˜C, so, by Theorem 2,BR GI ∩V is a clique ofH.

Therefore the maximal cardinality of a clique inH ∗ is the maximal cardinality of a

clique inH plus one and the second inequality is proved. ⊓⊔

10 Planar graphs which are not 3-connected

We have proved that, for any 3-connected planar graphG, the treewidth of its dual

is at most the treewidth ofGplus one. We extend this result to arbitrary planar

graphs.

The following lemma is a well-known result, see for example [12] for a proof:

Lemma 11. LetG = (V,E) be a graph (not necessarily planar) andS be a separator

ofGsuch thatG[S] is a complete graph. LetV 1 ,V 2 ⊆V such thatS,V 1 and

V 2 form a partition ofV andS separates each vertex ofV 1 from each vertex ofV 2 .

Then tw(G) = max(tw(G[S∪V 1 ]), tw(G[S∪V 2 ])).

Lemma 12. LetG=(V,E) be a graph, not necessarily planar. Suppose thatGhas

a minimal separatorS={x,y} of size two. LetG xy = (V,E∪{xy}) be the graph

obtained fromGby adding the edgexy. Then tw(G xy ) = tw(G).

Proof. Ifxy is an edge ofG, thenG xy =G. Suppose thatxy is not an edge ofG.

SinceGis a minor ofG xy , we have tw(G)≤tw(G xy ), so it remains to show that

tw(G)≥tw(G xy ).

S is also a minimal separator ofG ′ , so letC be a full component associated to

S inGand letV 2 =V\ (C∪S). LetG 1 =G xy [S∪C] andG 2 =G[S∪V 2 ]. By

Lemma 12, we have tw(G xy ) = max(tw(G 1 ), tw(G 2 )). It is sufficient to prove that

tw(G 1 )≤tw(G) and tw(G 2 )≤tw(G). We show thatG 1 andG 2 are minors ofG.

By Lemma 1, there is a full componentDassociated toS different fromC,

soD⊆V 2 . There is a pathµfromxtoyinG[D∪{x,y}], and the interior of

µ avoids the vertices ofG 1 . Therefore,G 1 is a minor ofG[S∪C∪µ], soG 1 is

a minor ofS. In a similar way, there is a pathµ ′ fromxtoyinG[C∪{x,y}],

soG 2 is a minor ofG[V 2 ∪S∪µ ′ ] and thus a minor ofG. We conclude that

tw(G)≥max(tw(G 1 ), tw(G 2 )) = tw(G xy ). ⊓⊔

Lemma 13. Suppose there is a plane graphGnot satisfying tw(G ∗ )≤tw(G) + 1

and there is a separatorS ={x,y} ofG. ThenG S also contradicts tw(G ∗ S )≤

tw(G S ) + 1.

Proof. By Lemma 12, tw(G S ) = tw(G). We also have thatG S is planar andG ∗ is

a minor ofG ∗ S . Indeed, ifxy is not an edge ofG, letC be a full component ofG\S

12


andν S (C) the cycle associated toS andC, close toC. Thenν S (C) = [x,f,y,f ′ ],

sox,y are incident to a same facef. We obtain a plane drawing ofG S by adding

the edgexy in the facef. The new edge will split the facef into two facesf 1

andf 2 , and clearly the dual ofGis obtained from the dual ofG S by contracting

the edgef 1 f 2 into a single vertexf. Therefore, tw(G ∗ )≤tw(G ∗ S ). Consequently, if

tw(G ∗ S )≤tw(G S) + 1, then tw(G ∗ )≤tw(G) + 1. ⊓⊔

Theorem 6. For any plane graphG,

tw(G ∗ )≤tw(G) + 1.

Proof. Suppose there is a graphGsuch that tw(G ∗ )>tw(G) + 1. We takeGwith

minimum number of vertices. It is easy to check thatGmust have at least four

vertices.

By Theorem 5,Gis not 3-connected, so letS be a minimal separator ofGwith

at most two vertices. According to Lemma 13, we can consider thatS is a clique

inG. LetC be a connected component ofG\S, we denoteG 1 =G[S∪C] and

G 2 =G[V\C] (ifGis not connected, thenS=∅ andC is a connected component

ofG). By Lemma 11, tw(G) = max(tw(G 1 ), tw(G 2 )).

The graphsG 1 andG 2 are clearly planar and they have less vertices thatG, so

tw(G ∗ 1)≤tw(G 1 ) + 1 and tw(G ∗ 2)≤tw(G 2 ) + 1. It remains to prove that tw(G ∗ )≤

max(tw(G ∗ 1 ), tw(G∗ 2 )).

Consider the case whenGis 2-connected. By proposition 3 there is a cycle

ν S (C) ofG I associated toS andC, close toC. The cycle contains four vertices,

˜ν S (C) = [x,f,y,f ′ ]. LetR 1 (respectivelyR 2 ) be the region ofΣ\ν S (C) containing

C (respectivelyV\ (S∪C)). Notice that the vertices ofG 1 (respectivelyG 2 ) are

exactly the original vertices of (˜ν S (C),R 1 ) (respectively (˜ν S (C),R 2 )). LetF 1 andF 2

be the face-vertices ofG I contained inR 1 , respectivelyR 2 . We denoteS f ={f,f ′ }.

LetG f 1 be the graph obtained fromG∗ [S f ∪F 1 ] by adding the edgeff ′ . Consider

the plane drawing ofG 1 obtained by restricting the drawing ofGat the one-block

region (˜ν S (C),R 1 ) and by adding the edgexy through the regionR 2 . It is easy to see

thatG f 1 is exactly the dual ofG 1. In a similar way, we define the graphG f 2 obtained

fromG ∗ [S f ∪F 2 ] by adding the edgeff ′ . If we consider the plane drawing ofG 2

obtained by restricting the drawing ofGto the one-block region (˜ν S (C),R 2 ) and by

adding the edgexy throughR 2 , thenG f 2 is the dual ofG 2. By the minimality ofG,

we have tw(G f 1 )≤tw(G 1) + 1 and tw(G f 2 )≤tw(G 2) + 1. Observe now that in the

graphG ∗ S f

obtained fromG ∗ by adding the edgexy,S f separatesF 1 fromF 2 . By

Lemma 11, tw(G ∗ S f

) = max(tw(G f 1 ), tw(G2 f )), so tw(G∗ S f

)≤max(tw(G 1 ), tw(G 2 ))+

1 = tw(G) + 1. We conclude that tw(G ∗ )≤tw(G) + 1.

The case whenGis not 2-connected is similar. Suppose thatGis connected

but no 2-connected, soS has a unique vertexx. There is a facef ofG I such that

we can draw a Jordan curve ˜ν S passing throughxandf, contained in the facef

(except the pointx), and the curve separatesC fromV\ (C∪{x})). IfGis not

connected, we can take a connected componentC and a facef such that a Jordan

curve ˜ν S contained in the facef, passing throughf, separatesC fromV\C. As

in the case of 2-connected graphs, we consider the regionsR 1 (respectivelyR 2 ) of

Σ\˜ν S containingC (respectivelyV\ (C∪S)). We takeS f ={f} and we denote

F 1 (respectivelyF 2 ) the face-vertices ofG I contained inR 1 (respectivelyR 2 ). Then

G f 1 =G∗ [S f ∪F 1 ] is the dual ofG 1 andG f 2 =G∗ [S f ∪F 2 ] is the dual ofG 2 .

We conclude that tw(G ∗ ) = max(tw(G f 1 ), tw(Gf 2 ))≤max(tw(G 1), tw(G 2 )) + 1, so

tw(G ∗ )≤tw(G) + 1.

⊓⊔

13


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